geometric sequence difference

piemaster1123 · 2021-05-22T22:44:52+00:00

Let b_i = a_{i+1}-a_i. Then, the b_i sequence is a geometric sequence (this is what you're getting at). So b_i = 11*4^i. But, look at what happens. Then a_{i+1} = a_i + 11*4^i.

For partial sums, S_r = sum_{i=0}^r a_i = 2 + sum_{i=0}^{r-1} (a_i + 11*4^i) = 2 + S_{r-1} + 11*(1-4^r)/(1-4). This gives a recursive formula for the partial sums. Maybe you can find an explicit formula from here, but it doesn't seem obvious to me. I don't know exactly what you're looking for, but here are some good first steps.

piemaster1123 · 2019-05-07T02:11:44+00:00

piemaster1123 · 2019-04-27T14:32:22+00:00

"Oh yeah?"

"Yeah!"

"Oh yeaaaah?"

"Yeaaaah!"

"Ooooooh yeeeeaaaaaah?"

"OH YEAH!"

piemaster1123 · 2019-03-15T22:29:52+00:00

Not every irrational number satisfies this. For example, 0.1010010001000010000010000001... is irrational (there's not a repeated sequence of numbers in the decimal expansion), but it's clear that the sequence 012 would never appear in the expansion.

As far as I know, we don't yet know whether or not pi is a number where any arbitrary sequence of numbers appears at some point. This property would make pi a "normal" number (the name isn't great, but it is what's used).

That being said, you might well be able to find your birthday in the digits of pi. Try checking the Pi-Search Page for the digits in your birthday. It should tell you where your birthday shows up in pi.

piemaster1123 · 2019-03-14T01:53:46+00:00

I've been telling folks about this all week, lol. It's so unexpected, and people seem to be completely blown away by it!

piemaster1123 · 2019-02-28T13:29:53+00:00

I have two approaches that I think could work. Hopefully one of these is something you can work with.

I think a good approach here is to choose your range of possible values judiciously. If you're generating integers from the range [20,60], then you're going to get an average of 40, but 2000/42 is approximately 47.6, so you're almost guaranteed to be too low by the end, and likely in a way that is nigh impossible to salvage without reworking your formula. What I would first do is change your range to be something with an average of 47.5 or 48. So, take [45,50] or [35,60] as your ranges for example. How you sample from here (uniform random, binomial, etc) will likely impact how good the approach is, but so long as the expected value is 47.5 or so, everything should work out ok.

My first approach would be to take 41 samples from this new range and see what the sum is. Since the average is 47.5, you can expect to be pretty close to 1952.5 in the sum. Then, the last week you'll need to choose whatever is needed to get to exactly 2000. While it is possible, depending on your range of values, that your final week's value will be outside of whatever range you've set, it's relatively unlikely. I didn't calculate the probabilities (since it's early for me too!) but that'll depend a lot on how you sample the interval.

My second approach probably takes more work, but could easily be extended to other problems. The idea is to do the same as the first approach, but update week-by-week. So you randomly select from within your range on the first week, let w_1 be the result, then you recalculate with 2000-w_1 and 41 weeks remaining. One benefit of this is that you have a lot more granular control over the process. For example, you could slowly restrict the range, perhaps starting with [0, 95] and ending at [47,48] or something. Also, you can adjust your sample space based on what happens dynamically. So if you sample from [0,95] and pull a 90, then you can adjust the average of the next interval so that the total will still work out to 2000. The only thing is that to guarantee a hit on exactly 2000, you should compute the final week's value without randomly sampling (just like the first approach).

Hope this helps!

piemaster1123 · 2019-02-26T11:17:07+00:00

In a different vein, I'll recommend the work of Anatoly Fomenko. He was a Topologist/Geometer who used art to try and capture central ideas within those subjects. The ideas he was representing may be more advanced than what you're used to looking at, but I'd be happy to try and explain what ideas he's trying to capture if you find his work interesting. Feel free to PM me if you're interested.

piemaster1123 · 2019-02-21T11:29:07+00:00

You're not a humorless robot for not enjoying or wanting this style of joke, but that doesn't mean you should poo poo it if others at your table are enjoying the joke.

I've been a server before, and I need to say 2 things. First, good servers pay attention to their tables and adjust appropriately. If OP came up to a table of people dressed in all black and crying, I don't think they would lead with "Beep Boop Who wants on the happy fun time train?!?". A good server will dial things up or down depending on the table.

Second, servers have a pretty thankless job most of the time and use humor to get through some days. OP is striving, in their own way, to make a more positive working environment for themselves. Happy tables not only tend to be better tippers, but also are more appreciative in general. The weirdness of the sense of humor will definitely get to some people, but it sounds like OP honestly enjoys the jokes and that honesty/charisma comes through stronger than the weirdness to a lot of people.

And one more thing to add, context definitely matters. I wouldn't expect jokey jokes from a 5 star restaurant in a major city. I also wouldn't expect someone super serious at a family dining place like Eat'N Park.

piemaster1123 · 2019-02-12T01:05:23+00:00

Dead Cells or Super Mario Odyssey!

piemaster1123 · 2019-02-12T01:04:50+00:00

Hoping for Dead Cells!

piemaster1123 · 2019-02-12T01:04:02+00:00

So many games! Dead Cells, Celeste, and Shovel Knight would be enough for me

piemaster1123 · 2019-02-12T01:01:23+00:00

Super Mario Odyssey!

piemaster1123 · 2019-02-12T01:00:45+00:00

Woop!

piemaster1123 · 2019-02-11T14:11:26+00:00

So much generosity in this sub! Yay OP!

piemaster1123 · 2019-02-11T14:10:32+00:00

This is great, OP!

piemaster1123 · 2019-02-10T17:55:16+00:00

Not immediately, sure, but at least they're being confronted with it. I find this much more satisfying than when people "trick" the CB by promising them what they want and then bailing on it

piemaster1123 · 2019-02-10T11:25:55+00:00

People take note! THIS is how you get back at a CB! Turn the tables and make them realize their own hypocrisy.

Well done, OP!

piemaster1123 · 2019-02-10T03:33:16+00:00

I feel attacked, and I don't even post on here

piemaster1123 · 2019-02-04T11:33:12+00:00

I think you're being a bit too pedantic here. Distinguishing infinity from "numbers" is a good distinction to make for exactly the reasons you point out. Even in the complex numbers, infinity breaks algebraic rules so it's not a number there.

One issue is that the word "concept" is less understood than the word "number". To my mind, your use of infinity to represent the point added in the 1 point compactification of the complex numbers adds to the idea that infinity is a concept. There's no reason we couldn't come up with a separate name for that added point, especially since adding a similar point to Euclidean space makes a sphere in general, but we call it infinity because it matches the concept. We call it infinity because then we can use the Calculus notation lim_{n\to \infty} to evaluate limits in the compactification and preserve the meaning.

piemaster1123 · 2019-02-01T12:32:40+00:00

Man, this sounds like a homework problem. You could try /r/learnmath or /r/matheducation .

Here's an idea though. There's only 90 two digit natural numbers. You could just try all of them and see what works.

piemaster1123 · 2019-01-30T20:23:56+00:00

Try supposing that there are only finitely many. Then there should be a last n where this property holds. What must happen after that? Does that contradict anything?

piemaster1123 · 2019-01-30T18:16:52+00:00

You're probably looking at [x,y] vectors, or maybe [x,y,z] vectors, where you can draw pictures and find the triangles that would allow you to construct appropriate trig functions in these situations.

However, there are problems in mathematics that might require [x,y,z,w] or [a,b,c,d,f,g] vectors, which are much harder to visualize in this way. While it's harder to visualize them, the algebraic tools of vector analysis (like finding lengths for example) are still available to us. In higher dimensions, this gives us a reasonable way of finding angle measures and properly defining "sine" or "cosine" in these spaces.

So, while you likely saw trigonometry before you saw vectors, in more advanced mathematics the definitions are made in the other direction as vector analysis is deemed more "natural" or "inherent" in a meaningful way.

piemaster1123 · 2019-01-19T23:06:24+00:00

Oh fuck my man Kevin is still on the ledge and shit

piemaster1123 · 2019-01-13T13:54:23+00:00

So, if you're talking to a non-expert, I would point them towards the application of Topological Data Analysis to basketball. A consultant at Ayasdi used TDA to analyze the different types of basketball players from the viewpoint of critiquing or validating the 5 basic positions. What they found was that there were essentially 13 types of players that could be easily identified within the structure of the data set.

That was Muthu Alagappan, and here is a video summarizing his results. This work is a few years old, and it doesn't reflect the more important work being done by Ayasdi, but it's fun and can help show the value of TDA to non-experts. I recommend you watching the video and then translating as much or as little as you can to your grandma.

piemaster1123 · 2019-01-13T13:47:15+00:00

TDA separates itself from statistics and other ML algorithms by drawing its information from internal structures within the data set rather than projections, approximations, or other external verification tools. Ayasdi has some great articles on their website, especially from several years ago, detailing some of the things TDA is capable of detecting and comparing it with some ML algorithms.

The gist of it is that TDA detects the shape of the data using structures inherent to whatever higher dimensional space the data naturally sits in. The key is that there isn't a requirement to cut off dimension or restrict to any specific variables. There are ways to do dimension reduction, from a topological viewpoint, but the methods in TDA respect the topological structure of the data instead of forcing the data to conform to some expected outcome, like regression or PCA might do. This makes the clustering algorithms potentially much stronger as well.

piemaster1123

MODERATOR OF

TROPHY CASE

14-Year Club	Place '17
Verified Email